The Yorkshire batsman was bidding to become the first Englishman to pass 50 in seven successive Test innings and, in the process, match a world record.
But having looked in sublime touch as his team-mates toiled in Bridgetown, he pushed at a wide delivery from spinner Veerasammy Permaul on 33 and got a thin edge to wicketkeeper Denesh Ramdin.
Cook, though, advanced to 60 not out at tea – a sixth half-century in nine Test innings for a player whose perceived lack of form has been discussed at length – and was building a useful partnership with Moeen Ali.
Only 25 overs were bowled during a ponderous opening session at the Kensington Oval in Bridgetown but that was enough for the hosts to dismiss both Jonathan Trott and Ian Bell for ducks as well as Gary Ballance for 18.
Cook won the toss and elected to bat but West Indies were given early impetus when Shannon Gabriel extended Trott’s miserable return to Test cricket by removing the opener in the second over.
Trott had retained his place in the side despite making just one score of note in the first two matches – 59 in the first innings at St George’s – but Gabriel’s short and fast delivery exposed familiar weaknesses as he spooned the ball up to square leg where Permaul took an easy catch.
Cook took advantage of two short balls from Jerome Taylor, swatting both to the leg-side boundary, before Jason Holder and Marlon Samuels were introduced to the home side’s attack.
Ballance hit the first six of the day in the 15th over, hooking Samuels high over mid-wicket, but the in-form left-hander’s innings came to an abrupt end five balls later when Holder removed his middle stump with a fizzing inswinger.
That brought Bell to the crease at 38 for two and the experienced number four lasted just five balls before chipping a return catch to Holder.
Under pressure, England were counting on the in-form Root and he whipped Holder’s full delivery off his legs to score his first boundary of the day, looking untroubled before lunch.
Cook also seemed to be finding his rhythm, although the England captain was made to sweat when he clipped to short leg, but umpire Billy Bowden judged the ball to have hit the ground before the catch was taken by Jermaine Blackwood.
Root moved past 30 with two swept boundaries in the space of three Permaul deliveries but the left-armer struck the decisive blow in their battle to bring Moeen to the wicket.
The Worcestershire all-rounder brought England’s 100 up with an off-balance, but cleanly-struck, four over long-on.
Home captain Ramdin wasted a review on an lbw appeal against Moeen, who was clearly struck outside the line by Samuels’ delivery.
England were 115 for four at the afternoon drinks break, with Cook on 49, and he steered Samuels past slip for two to reach a third successive half-century. It came in 139 balls and featured seven boundaries. A huge caught-behind appeal against Cook was turned down, the ball from Gabriel maybe flicking the top of the left-hander’s pad. His stand with Moeen, who had established himself nicely on 37, had advanced to 60 when tea arrived with England 151 for four.